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(B) Let the velocity of pulley $P_2$ be $v_{P2}$. The velocity of the center of a pulley is the average of the velocities of the two ends of the strin

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$4$

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(B) Let the velocity of pulley $P_2$ be $v_{P2}$. The velocity of the center of a pulley is the average of the velocities of the two ends of the string passing over it.For pulley $P_2$,the velocities of blocks $C$ and $D$ are $v_C$ and $v_D$ respectively. Let the upward direction be positive.Given $v_D = -4 \ m/s$ (downward).Since the top pulley is fixed,the velocity of the string connected to pulley $P_2$ must be equal and opposite to the velocity of the string connected to pulley $P_1$. From the left side,block $A$ moves down at $6 \ m/s$ and $B$ moves up at $6 \ m/s$,so the velocity of pulley $P_1$ is $v_{P1} = \frac{(-6) + 6}{2} = 0 \ m/s$.Since the string connects $P_1$ and $P_2$ over a fixed pulley,the velocity of $P_2$ must be $v_{P2} = -v_{P1} = 0 \ m/s$.For pulley $P_2$,$v_{P2} = \frac{v_C + v_D}{2}$.Substituting the values: $0 = \frac{v_C + (-4)}{2}$.Therefore,$v_C = 4 \ m/s$ (upward).

In the figure shown,the velocities of different blocks are given. The velocity of $C$ is ......... $m/s$.

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